RATIONAL EXPECTATIONS AND THE COURNOT-THEOCHARIS PROBLEM

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1 RATIONAL EXPECTATIONS AND THE COURNOT-THEOCHARIS PROBLEM TÖNU PUU Received 18 April 006; Accepted 1 May 006 In dynamic models in economics, often rational expectations are assumed. These are meant to show that the agents can correctly foresee the result of their own and the other agents actions. In this paper, it is shown that this cannot happen in a simple oligopoly model with a linear demand function and constant marginal costs. Naive expectations, that is, where each agent assumes the other agents to retain their previous period action, are shown to result in a -period cycle. However, adapting to the oserved periodicity always doules the actual resulting periodicity. In general, it is impossile for the agents to learn any periodicity except the trivial case of a fixed point. This makes the whole idea of rational expectations untenale in Cournot oligopoly models. Copyright 006 Tönu Puu. This is an open access article distriuted under the Creative Commons Attriution License, which permits unrestricted use, distriution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Ina notein1959, Theocharis [5] reconsideredthecournot [] oligopoly prolem, given a linear demand function and constant marginal costs for the competitors. He showed that with three competitors, the Cournot equilirium ecomes neutrally stale, and with four, it ecomes unstale. As discussed in [1], the whole argument was antedated 0 years earlier y Palander in [3, 4], ut as this seems to e totally unknown to the economics profession, we continue referring to the Theocharis prolem. However, oth Palander and Theocharis only considered loss of local staility of the Cournot equilirium point. In [1], the gloal dynamics of the model was studied. A linear demand function and constant marginal costs lead to linear reaction functions, which in terms of sustance, ecome nonsense once they result in negative output and profits that are positive due to negative costs dominating over negative revenues. Using nonnegativity constraints, the model necessarily ecomes nonlinear, which is needed for the study of the gloal dynamics. Hindawi Pulishing Corporation Discrete Dynamics in Nature and Society Volume 006, Article ID 3103, Pages 1 11 DOI /DDNS/006/3103

2 Rational expectations and the Cournot-Theocharis prolem The main result of [1] was that the model could asymptotically go to a stale monopoly, a stale duopoly, or else a synchronized -period oscillation, in the case of more than three competitors, one where all firms drop out every second period. The outcome solely depended on the marginal costs of the competing firms. After restating some of these facts, we will now consider that a -period cycle (resulting from so-called naive expectations) is easily learnt y the competitors. In particular, if all firms drop out every second period, any firm may attempt to ecome a monopolist these very periods, and so improve its long-run profit situation. According to the rational expectations hypothesis, all the competitors would learn this -periodicity, and so react to conditions, not in the past period, ut periods ack. As we will see, this changes the resulting periodicity, the cycle ecomes 4-periodic. But then, again assuming that all the competitors learnt this, we find that the resulting cycle is douled, to 8-period, and so forth. The general issue then is whether it is at all possile for the competitors to learn the right periodicity resulting from their own and their competitors actions, in accordance with the rational expectations hypothesis. It will e shown that no periodic orit, except an equilirium, can e learnt y the agents. In a very vague sense, of course, this reminds of the uncertainty principle in physics. By attempting to measure something, the oject of measurement is affected so as to make accurate oservation impossile. In the same way, learning the outcome of a dynamical process y the agents involved and adapting to it alters the process so that accurate learning of the process ecomes impossile.. Palander (1936/1939) and Theocharis (1959).1. The model. To restate the asic facts of the gloal dynamics, let us now specify the model. With linear demand, market price i=n p := a q i (.1) depends on the total of the supplies of the n competitors. To simplify notation, define total supply i=1 i=n Q := q i, (.) and residual supply (not under the control of the ith competitor) i=1 The revenue for the ith competitor hence ecomes Q i := Q q i, i = 1,...,n. (.3) R i = ( a ( Q i + q i )) qi, i = 1,...,n, (.4)

3 Tönu Puu 3 whence marginal revenue ecomes Given constant marginal costs, total costs are and marginal costs are R i q i = a Q i q i, i = 1,...,n. (.5) C i = c i q i, i = 1,...,n, (.6) C i q i = c i, i = 1,...,n. (.7) Equating marginal revenue (.5) to marginal cost (.7), we can solve for q i : q i = 1 a c i 1 Q i, i = 1,...,n, (.8) which can e considered either as a simultaneous system or as a recurrence system... Cournot equilirium. Inthefirstalternative,(.8), given definitions (.)and(.3), can e solved for the Cournot equilirium point: where for convenience q i = 1 a c i + n n +1 n +1 Summation of (.9) gives total output in Cournot equilirium: i=n c c i, i = 1,...,n, (.9) c = 1 c i. (.10) n Q = i=1 n a c n +1. (.11) We see that total Cournot equilirium output is a fraction of the ratio of maximum price minus average marginal cost to the slope of the inverse demand function. The fraction is one half in the case of monopoly, two thirds in the case of duopoly, and it goes asymptotically to unity as the numer of competitors increases. From (.11), we can also easily calculate price in the Cournot point: p = a Q = 1 n +1 a + n c, (.1) n +1 which is a weighted average of maximum price and average marginal cost, and hence

4 4 Rational expectations and the Cournot-Theocharis prolem positive. Using (.1)with(.9), the profits of the individual firms are Π i = ( ( ) 1 a c p c i qi = i + n n +1 n +1 ) c c i. (.13) This squared right-hand side may seem to e always nonnegative, ut in case the expression in rackets is negative, we see from (.9) thatq i < 0, which is nonsense. The positive profits then result from negative costs dominating over negative revenues. We will meet this consideration again when constructing the iteration for gloal dynamics. Hence, in order that a given firm stays in usiness, we require that (1/(n + 1))((a c i )/)+(n/(n + 1))((c c i )/) > 0. As the common denominators (n +1)>0, this can e reorganized as follows: c i < 1 n +1 a + n c. (.14) n +1 Only firms with sufficiently low marginal costs will actually stay active in a Cournot equilirium. Note that from (.10), c i is one of the terms in nc = i=n i=1 c i,soinordertostatea condition for c i in terms of all the c j, j i, we later need a further reorganization..3. Reaction functions. As mentioned, we may want to treat (.8) asadynamicalsystem. Suppose each firm assumes all the competitors to retain their supplies from the previous period. Then the est reply of this firm is q i (t +1)= 1 a c i 1 Q i(t), i = 1,...,n. (.15) This function (.15)isusuallycalledthereaction function. In this case, it is linear, due to the underlying linearity of the demand and cost functions, ut in general it is not, and then it might e impossile to state in explicit form. As we will see, the reaction function turns nonlinear (piecewise linear) also when we take nonnegativity of output and profits in account. Note that the derivatives of (.15), q i (t +1) Q i (t) = 1, i = 1,...,n, (.16) are constant, equal to 1/, independent of the coordinates of the Cournot equilirium point. For further use, take the sum of (.15)overi,anduse(.), (.3), and (.10)tootain Q(t +1)= n 1 a c Note that this is a divergent process whenever n>3. n 1 Q(t). (.17)

5 Tönu Puu 5.4. Local staility. The derivatives (.16) enter the Jacoian matrix of the system in all off-diagonal elements, whereas the diagonal elements are zero. Thus, The characteristic equation reads J =. (.18) ( J λi = λ 1 ) n 1 ( λ + n 1 ) = 0. (.19) There are n 1eigenvaluesλ 1,...,n 1 = 1/ and1eigenvalueλ n = (n 1)/. Theocharis concluded that in duopoly, there are two eigenvalues λ 1 = 1/ andλ = 1/, oth less than unity in asolute value, so the Cournot equilirium is stale. In triopoly, the eigenvalues are λ 1, = 1/, λ 3 = 1, so the equilirium is neutrally stale, with a tendency to endless ut stationary oscillation. For n>3, λ n =(n 1)/ > 1, so the equilirium ecomes unstale. Palander stated the same suject 0 years earlier, though for more general reaction functions. In his 1939 article, Palander [3] wrote as a condition for an equilirium with a certain numer of competitors to e stale to exogenous disturances, one can stipulate that the derivative of the reaction function f must e such that the condition (n 1) f < 1 holds. If this criterion is applied to, for instance, the case with a linear demand function and constant marginal costs, the equiliria ecome unstale as soon as the numer of competitors exceeds three. Not even in the case of three competitors will equilirium e restored, rather there remains an endless oscillation. As we see, Palander considers not just linear demand (and hence reaction) functions, ut staility of the Cournot equilirium in general. As usual, the eigenvalues correspond to a diagonalization of the system, the first n 1 associated with differences of the phase variales, the last with the sum of the phase variales. 3. The gloal dynamics 3.1. Nonnegativity of output and profits. It is now time to consider that q i (t + 1) cannot ecome negative. According to (.15), they must e restricted to Q i (t) a c i, i = 1,...,n. (3.1) However, there is another fact that we must take in account! Profits must e positive, or the firms will drop out from the market. So, consider expected profits as otained from

6 6 Rational expectations and the Cournot-Theocharis prolem (.4)and(.6), Π i (t +1)= ( a c i ( Q i (t)+q i (t +1) )) q i (t +1), i = 1,...,n. (3.) Sustituting for q i (t +1)from(.15), we get Π i (t +1)= ( a ci Q i (t)), i = 1,...,n. (3.3) 4 The right-hand side of (3.3) eing squared, it might seem that we always have a nonnegative profit. However, if (3.1) is not fulfilled, the nonnegativity of profits is due to positive costs dominating over negative revenues, a meaningless case in terms of sustance. Hence, the condition for a meaningful nonnegative profit coincides with the condition for nonnegative output, which actually simplifies things. 3.. The gloal reaction function. Hence, we reformulate the map as follows: 1 a c i 1 Q i(t), Q i (t) a c i, q i (t +1)= 0, Q i (t) > a c i = 1,...,n. (3.4) i, Numerical experiments with (3.4) indicate that eventually the model can go to steady states of monopoly or duopoly. With three competitors, a permanent oscillation appears, and with four or more, oscillations of period two, where all firms drop out every second period, are the rule. However, any numer of competitors may drop out, and the remaining firm(s) may go to Cournot duopoly or even a monopoly state. 4. Monopoly and duopoly We saw that (i) monopoly and duopoly are always stale, and (ii) it is not certain that all the n firms can make positive profits in the Cournot equilirium point. It may even occur that the relevant condition (.14), dependent only on the relation etween marginal costs, is fulfilled for only one or two firms. If so, the system settles to a stale monopoly or duopoly. The triopoly is a oundary case. We will first check out the possiility for monopoly and duopoly, and then, skipping the oundary case, estalish the general gloal dynamics for n>3 remainingcompetitors Monopoly. In the case of monopoly, where we, without loss of generality, can take the monopolist as numer 1, (.14)statesthatif c i > 1 n +1 a + n c, i =,...,n, (4.1) n +1 holds, then no competing firm will e ale to enter. We can reduce the question to first deal with excluding the possiility of any other firm entering into duopoly. Then n =, and we compare firms 1 and, where the second firm, again without loss of generality,

7 Tönu Puu 7 can represent any of the other firms. We thus get c > 1 3 a + 1 ( ) c1 + c 3 (4.) or, rearranging, c > 1 ( a + c1 ). (4.3) If this condition does not hold, then the second firm can enter into competition with the monopolist and estalish a duopoly. In order for a new firm to make positive profit, and enter in competition with the monopoly, its marginal cost must not exceed a certain numer, ut note that it does not need to e lower than the marginal cost for the monopolist. If a = 1, c 1 = 0.5, then the monopoly remains when c > 0.75, ut transforms into a duopoly when c < 0.75 even though the marginal cost of the second firm is 50% higher than that for the monopolist. As the Cournot point in duopoly is contractive, with eigenvalues ±1/, the system will approach the Cournot equilirium. 4.. Duopoly. Suppose now that one competitor succeeded to intrude the market, and that a duopoly is formed. Again c i > 1 n +1 a + n c, i = 3,...,n (4.4) n +1 must hold for the duopoly to remain a duopoly. We can again discuss a possile intrusion of a third firm, i = 3, as a generic representative for the remaining firms. Thus c 3 > 1 4 a + 1 ( ) c1 + c + c 3 (4.5) 4 prevents intrusion. Rearranging, c 3 > 1 3 ( a + c1 + c ). (4.6) Suppose again a = 1, c 1 = 0.5, and further c = 0.6. Then the nonintrusion condition reads c 3 > 0.7, whereas if c 3 < 0.7, the third firm will enter. As the eigenvalues are now 1/, 1/, 1, the system will ecome oscillating. However, as illustrated in [1], the new firm can also expel one of the duopolists, so that duopoly remains. This can also happen in the case of monopoly, where a new competitor can throw the previous monopolist out from the market. 5. The oscillatory case 5.1. Variale differences. Consider now the general oscillatory case with n>3. First let us form the difference (q j (t +1) q i (t + 1)) according to (.15), also taking note of (.) and (.3). Hence, ( q j (t +1) q i (t +1) ) = 1 c i c j + 1 ( q j (t) q i (t) ). (5.1)

8 8 Rational expectations and the Cournot-Theocharis prolem As we see, this is a contraction, so all differences etween variales eventually approach fixed quantities. This relates to what was said aove aout the first n 1 eigenvalues eing associated with differences of phase variales. 5.. The asymptotic process. After any transient has een passed, we have (q j (t) q i (t))= (c i c j )/,oretter, q j (t) = q i (t)+ c i c j. (5.) Summing over j, using (.)and(.10), we get or, solving for q i (t), Q(t) = nq i (t)+n c i c (5.3) q i (t) = c c i + 1 Q(t), i = 1,...,n. (5.4) n This tells us how each phase variale asymptotically changes with total output of the industry. For the dynamics of Q(t), we have (.17), which as we saw, unlike the case of (5.1), is oscillatory divergent whenever n>3. But this would sooner or later result in anegativeq(t), which cannot happen unless some q i (t) are negative as well, and this cannot happen. As we are past the transient, (5.4) holds. Further, assume Q(t) = 0. Then from (5.4), q i (t) = c c i, i = 1,...,n, (5.5) we conclude that q i (t) > 0forc i < c, whereasq i (t) < 0forc i > c. The latter is, however, impossile, so the negative outputs are replaced y zero. If there are any positive outputs, then their aggregate must e positive too, ecause they cannot e counteralanced y negative ones, that is, Q(t) > 0, which contradicts that Q(t) = 0. Thus, in the asymptotic state, all q i (t) = 0wheneverQ(t) = The -period orit. Assume now that Q(t) = 0insometimeperiod.From(.17), we then find Q(t +1)= n 1 a c > 0, (5.6) where positivity holds ecause maximum price must e higher than all marginal costs, and hence also their average. Further consider also the following iterate, otained through sustituting (5.6)in(.17) once more, whence Q(t +)= n(n 3) 1 a c < 0 (5.7) 4 negative as stated provided that n>3.

9 Tönu Puu 9 This cannot happen, ecause Q(t +)< 0 implies that some q i (t +)< 0. Any negative q i (t + ) are, however, replaced y zero. Then, positive q i (t + ) cannot occur, ecause there are no negative ones that can counteralance them to make Q(t +)< 0. Hence, all q i (t +)= 0, which also sum up to Q(t +)= 0. We conclude that (i) zero total market output after the passage of two periods again results in a zero output, and (ii) this can only happen through all individual outputs eing zero simultaneously as well. From (5.6), we find that the same positive total quantity is produced in the intermediate periods. From (5.4), asymptotically there is a fixed linkage etween total output and all the individual outputs, so there definitely is a -period cycle where all the firms drop out every second period. This also shows up in any numerical experiment. The condition for this type of -period cycle to occur is that the remaining numer of firms is n>3 as assumed. 6. Learning the orit 6.1. Period douling. A -period orit is the simplest possile nonconstant orit, so as soon as the competitors find this out, everyody will adapt to conditions two periods ack. They know that their first expectations that the competitors retain their supplies from the previous period turned out wrong, and they learned that the competitors actions recur every second period. Given this, we can immediately state the recurrence equation 1 a c i 1 Q i(t), Q i (t) a c i, q i (t +)= 0, Q i (t) > a c i = 1,...,n. (6.1) i, Note that nothing is changed, except the adaptation period. The Cournot equilirium point is the same, and all considerations aout entry and positivity of profits still hold. Recall that we are dealing with n>3, that is, marginal costs such that all competitors remain in usiness. Oviously, assuming q i (t) = Q(t) = Q i (t) = 0 and using the aove considerations, we conclude that now q i (t +4)= Q(t +4)= Q i (t +4)= 0 as well. In the intermediate period t +, all quantities will e positive. So much aout periods t, t +, and t +4. For periods t +1, t +3, and t +5, there is another independent 4-period cycle, with alternating zero and positive outputs. How these two processes are comined depends on initial conditions and transients. Anyhow, the resulting process is 4-period, so again, the assumption that the resulting cycle was -periodic shows up to e wrong. We can further assume that the competitors learn also this 4-periodicity, only to check out that they actually produce an 8-period cycle, and so it goes. 6.. Possiility of rational expectations. We may thus pose the following question. Does there exist any periodicity which the competitors can learn, and which coincides with what their ehaviour actually results in? Our answer is in the negative.

10 10 Rational expectations and the Cournot-Theocharis prolem Assume there is such a T-periodicity. Then, in analogy to (3.4)and(6.1), we have 1 a c i 1 Q i(t), Q i (t) a c i, q i (t + T) = 0, Q i (t) > a c i = 1,...,n. (6.) i, But under the rational expectations hypothesis, the T-periodicity is recognized, so q i (t + T) q i (t). (6.3) Sustituting from (6.3)in(6.) and taking the first ranch of the map, or, using (.3), Finally, rearranging, q i (t) = 1 a c i 1 Q i(t) (6.4) q i (t) = 1 a c i 1 Q(t)+1 q i(t). (6.5) q i (t) = a c i Sum over i,use(.)and(.10), and rearrange to solve for Q(t) = Q(t). (6.6) n a c n +1, (6.7) which equals total supply in Cournot equilirium, that is, Q(t) = Q. Further, sustituting ack in the previous equation, we get q i (t) = 1 a c i + n n +1 n +1 c c i, i = 1,...,n, (6.8) which equals the individual supplies in Cournot equilirium, that is, q i (t) = q i, i = 1,...,n. (6.9) As these relations have to hold for all t, we conclude that the only periodic orit that can e learned is the fixed Cournot point. Now, we chose the first ranch of the reaction function (6.), so how aout the second? From the aove reasoning around (.13), we know that output and profits in Cournot equilirium are positive, so it is this ranch that counts. By conclusion, in terms of rational expectations, the fixed point is the only periodic orit that can e learned y the agents.

11 7. Discussion Tönu Puu 11 The Cournot oligopoly model with linear demand and constant marginal costs was shown y Palander [3] andtheocharis[5] to have an unstale equilirium whenever the numer of competitors ecomes larger than 3. The gloal dynamics of this model can only e studied if nonnegativity constraints for supplies and profits are taken in account. Given this is done, the model displays a -period orit when the competitors react to naive expectations, that is, assume that everyody else will supply as in the previous period. Oviously these expectations are not fulfilled. If the competitors learn the -periodicity, and adapt their eahviour to it, the actual periodicity is douled to 4, and this goes on with any expectations that try to capture the true periodicity. Learning a periodicity interferes with the process, and alters it. By conclusion, it ecomes ovious that the only periodicity that can e learned in the rational way is the Cournot fixed point. But as we saw, this fixed point ecomes unstale, quite as pointed out y Palander and Theocharis, so it is no alternative with more than 3 competitors. The natural question then is whether the competitors can learn and adapt to anything else than a periodic orit. A dynamical system can only e attracted to fixed points or periodic orits, to quasiperiodic or chaotic orits, or it can diverge. Now, in the oligopoly case, divergence indicates misspecification of the model, for instance y neglecting nonnegativity constraints in a linear model. On the other hand, a chaotic trajectory is in principle unpredictale, and a quasiperiodic one requires infinite memory. So, the rational expectations hypothesis seems impossile. It is the conviction of the author that this holds in much more generality, in fact for any reasonale oligopoly model we may construct, and perhaps also in other classes of dynamical models in economics. References [1] J. Cánovas, T. Puu, and M. Ruiz, The Cournot-theocharis prolem revisited, to appear in Chaos, Solutions, and Fractals. [] A. Cournot, Recherches sur les Principes Mathématiques de la Théorie des Richesses, Hachette, Paris, [3] T. F. Palander, Konkurrens och marknadsjämvikt vid duopol och oligopol, Ekonomisk Tidskrift 41 (1939), , 50. [4], Instaility in competition etween two sellers, Astracts of Papers Presented at the Research Conference on Economics and Statistics held y the Cowles Commission at Colorado College, Colorado College, General Series no. 08, Studies Series no. 1, [5] R. D. Theocharis, On the staility of the Cournot solution on the oligopoly prolem, Review of Economic Studies 7 (1959), Tönu Puu: The Center for Regional Science (CERUM), Umeå University, Umeå 90187, Sweden address: tonu.puu@econ.umu.se